Partition-Symmetrical Entropy Functions

Let $\cal{N}=\{1,\cdots,n\}$. The entropy function $\bf h$ of a set of $n$ discrete random variables $\{X_i:i\in\cal N\}$ is a $2^n$-dimensional vector whose entries are ${\bf{h}}({\cal{A}})\triangleq H(X_{\cal{A}}),\cal{A}\subset{\cal N}$, the (joint) entropies of the subsets of the set of $n$ random variables with $H(X_\emptyset)=0$ by convention. The set of all entropy functions for $n$ discrete random variables, denoted by $\Gamma^*_n$, is called the entropy function region for $n$. Characterization of $\Gamma^*_n$ and its closure $\overline{\Gamma^*_n}$ are well-known open problems in information theory. They are important not only because they play key roles in information theory problems but also they are related to other subjects in mathematics and physics. In this paper, we consider \emph{partition-symmetrical entropy functions}. Let $p=\{\cal{N}_1,\cdots, \cal{N}_t\}$ be a $t$-partition of $\cal N$. An entropy function $\bf h$ is called $p$-symmetrical if for all ${\cal A},{\cal B} \subset {\cal N}$, $\bf{h}({\cal A}) = \bf{h}({\cal B})$ whenever $|{\cal A} \cap {\cal N}_i| = |{\cal B} \cap {\cal N}_i|$, $i = 1, \cdots,t$. The set of all the $p$-symmetrical entropy functions, denoted by $\Psi^*_p$, is called $p$-symmetrical entropy function region. We prove that $\overline{\Psi^*_p}$, the closure of $\Psi^*_p$, is completely characterized by Shannon-type information inequalities if and only if $p$ is the $1$-partition or a $2$-partition with one of its blocks being a singleton. The characterization of the partition-symmetrical entropy functions can be useful for solving some information theory and related problems where symmetry exists in the structure of the problems. Keywords: entropy, entropy function, information inequality, polymatroid.Comment: This paper is published in IEEE Transactions on Information Theor

Boundary Integral Equations for the Laplace-Beltrami Operator

We present a boundary integral method, and an accompanying boundary element discretization, for solving boundary-value problems for the Laplace-Beltrami operator on the surface of the unit sphere $\S$ in $\mathbb{R}^3$. We consider a closed curve ${\cal C}$ on ${\cal S}$ which divides ${\cal S}$ into two parts ${\cal S}_1$ and ${\cal S}_2$. In particular, ${\cal C} = \partial {\cal S}_1$ is the boundary curve of ${\cal S}_1$. We are interested in solving a boundary value problem for the Laplace-Beltrami operator in $\S_2$, with boundary data prescribed on \C

Controlling quantum entanglement through photocounts

We present a protocol to generate and control quantum entanglement between the states of two subsystems (the system ${\cal S}$) by making measurements on a third subsystem (the monitor ${\cal M}$), interacting with ${\cal S}$. For the sake of comparison we consider first an ideal, or instantaneous projective measurement, as postulated by von Neumann. Then we compare it with the more realistic or generalized measurement procedure based on photocounting on ${\cal M}$. Further we consider that the interaction term (between ${\cal S}$ and ${\cal M}$) contains a quantum nondemolition variable of ${\cal S}$ and discuss the possibility and limitations for reconstructing the initial state of ${\cal S}$ from information acquired by photocounting on ${\cal M}$.Comment: 12 pages, 3 figures, accepted for publication in Phys. Rev